Given φ ₁ ,φ ₂ ∈ L¹ ([0,T]) and a function x ∈ W2,1 ([0,T]) solving the control problem (P) x'' + a₁ (t)x' + a₀ (t)x ∈ [φ ₁ (t),φ ₂ (t)] a.e., x(0) = x₀, x(T) = x₁, x'(0) = v₀, x'(T) = v₁ there exists a bang-bang solution y to (P) satisfying y x; moreover there exists a finite union of intervals E such that y'' + a₁ y' + a₀ y = φ ₁ χ E + φ ₂ χ [0,T] E. The reachable set of bang-bang constrained solutions is convex: an application to the calculus of variations.
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Cerf et al. (1995) studied this question.